Fungal spores are all around us. Fungi are how dead stuff is broken down. Most are beneficial, many are harmless, very few are harmful. The fungus is unlikely to have killed your tree it will have moved into dead wood.

Some turn quicker, yes. They all disappear when the plants leaf up in spring though. All adds to the organic stuff in the soil. I would remove them from around tiny plants or those that like it well drained and airy, but otherwise leaves stay where they fall.